Nicolai Map Research Articles

Topological field theories, Nicolai maps and BRST quantization

Abstract We establish a connection between topological field theories, Nicolai maps, BRST quantization and Langevin equations. In particular we show that there is a one-to-one correspondence between global unbroken supersymmetric theories which admit a Nicolai map and theories which arise as the BRST quantization of the square of the Langevin equation, setting the random field to zero. As such they are topological in nature. As an example we consider the topological quantum field theory of Witten in the Labastida-Pernici form and show that it is the first example of a theory admitting a complete Nicolai map in four dimensions. We also consider the topological sigma models of Witten and show that they too arise from the BRST quantization of the square of the Langevin equation.

Linear lattice gauge theory with Nicolai mapping.

We construct a lattice gauge theory with linear gauge variables and fermions, which has an explicit Nicolai mapping and is invariant under a supersymmetric translation \ensuremath{\theta}\ensuremath{\rightarrow}\ensuremath{\theta}+\ensuremath{\epsilon}. It contains a new interaction which removes the pathology of Weingarten's model for Nambu-Goto random surfaces. Fermions generate curvature interactions. Strong-coupling expansion and supersymmetry breaking are discussed.

Stochastic representation of non-abelian supersymmetric quantum mechanics

Two stochastic representations of two-dimensional non-abelian supersymmetric magnetic field systems are studied. In the case of SU(2) symmetry, the first is written in terms of a Markov process with continuous component coupled to a discrete component, provided a positivity condition is satisfied. The second is based upon an embedded U(1) model and leads to a representation of the euclidean wave function as a stochastic average with respect to gaussian white noise. The relation with the Nicolai map for the abelian model is discussed.

On the Nicolai map and Witten index for two-dimensional supersymmetric magnetic field systems

Various aspects of two-dimensional supersymmetric quantum mechanical systems with magnetic fields are studied. In particular, Nicolai maps are obtained and an associated stochastic interpretation in terms of a Markovian process with non-potential forces is given. The evaluation of the Witten index in both the heat kernel and resolvent regularisation scheme is considered. An explicit calculation of a specific model is presented, clarifying how the non-integer value of the index is built up.

The degree of the Nicolai map in supersymmetric quantum mechanics

In an earlier paper, we developed a degree theory for Wiener maps [O]. In this paper, we will give an application of this theory which arises in the study of path integrals for supersymmetric Hamiltonians in quantum mechanics. Before defining the map which we will be looking at, we must specify the Wiener space on which it acts. This is the space of HGlder loops C’(.S, R”‘), c( < 4, with the Ornstein-Uhlenbeck measure d;l-the Gaussian measure corresponding to the Hilbert subspace L’x’(S, R”“), which is the Hilbert space of functions on the circle for which the inner product

Observations on the Nicolai map at finite temperature.

We show that the Nicolai map in supersymmetric theories breaks down at finite temperature emphasizing that supersymmetry is broken for Tnot =0.

Stochastic representation of arbitrary fermion sectors in supersymmetric quantum mechanics on Riemannian manifolds.

In the zero-fermion and n-fermion sectors of Euclidean supersymmetric quantum theories on an n-dimensional manifold a transformation is known which reduces the functional integrals to Gaussian functional integrals. The transformation, often called a Nicolai map, can be interpreted as a Markovian stochastic process. We investigate two different possibilities to generalize this result to an arbitrary p-fermion sector (0\ensuremath{\le}p\ensuremath{\le}n), and find that only one of them is viable, in general. We give a general representation of the solution of the Schr\"odinger equation in the p-fermion sector in terms of a Markovian stochastic process which extends the stochastic process of the zero- and n-fermion sectors.

A simple Grassmannian path integral representation of the Dirac propagator

Starting from the affinely parametrized supersymmetric Dirac particle model a Grassmannian path integral expression for the propagator of the Dirac equation, minimally coupled to an external electromagnetic field, is derived. A purely ‘‘bosonic’’ path integral representation of the propagator of the iterated minimally coupled Dirac equation is also obtained. It appears that a Nicolai mapping exists, even in a formal sense, only in the case of a constant external field.

Regularization and noncovariant gauges

We examine the advantages and disadvantages of noncovariant gauges, with special emphasis on the light-cone gauge. We describe the correct iε prescription in this gauge and discuss a number of other technical aspects. The role of the light-cone gauge in connection with Nicolai maps and stochastic quantization is also examined.

Nicolai maps and fermion sectors

Nicolai maps, and particularly their fermion sector structure, are explored over a broad variety of supersymmetric theories. Studies are made in the Witten model, the N = 2 supersymmetric sigma model in 1 + 0 dimensions, dimensionally reduced Wess-Zumino models, and N = 1 Yang-Mills in D = 4. In general each theory yields families of Nicolai maps, each map being a Langevin equation, equivalent to the dynamics of a particular sector. We also detail the remarkable fact that closed form Nicolai maps exist for a few ordinary bosonic nonsupersymmetric theories, with ordinary QCD 4 as an important example. These bosonic theories correspond to “null-fermion” sectors of supersymmetric theories, in which the fermions fail to contribute.

Fermionic path integrals, the nicolai map and the witten index

We present an explicit evaluation of the coherent-state fermion path integral and discuss our results in the light of supersymmetric quantum mechanics, the Nicolai map and the Witten index.

On the 2-dimensionalN=2 Wess-Zumino model on the lattice; From Dirac-K�hler formalism to the Nicolai mapping

We present a treatment of the supersymmetric Wess-ZuminoN=2 two dimensional model in the discrete space-time. We propose a pure differential form approach to the superspace formalism which enables us to obtain the corresponding supersymmetric Dirac-Kahler Lagrangean. We show that our discretized supersymmetric model can be alternatively constructed from Nicolai mapping on the lattice.

A perturbative study of the nicolai mapping

We show that in N = 1 supersymmetric Yang-Mills theory up to one loop, two- and three-point gluon Green's functions computed in the light-cone gauge with the Mandelstam prescription by inverting the Nicolai mapping coincide with the results obtained in ordibary perturbation theory. In the covariant gauge a discrepancy appears in the case of the three-point function.

Witten's index and supersymmetric quantum mechanics

Criteria for the spontaneous breaking of supersymmetry have been proposed and we examine their range of applicability by evaluating them in some quantum mechanical models. In particular, Str (e−βH) as well as the functional integral expression by Cecotti and Girardello may differ from the integer Δ obtained by zero energy state counting if the spectrum has a continuum part. An attempt to elucidate the difference is made. Also the Nicolai mapping may not have complete range even if the symmetry is unbroken.

Gauge dependence of the Nicolai map in super Yang-Mills theory

Four-dimensional super Yang-Mills theory is reformulated in terms of the self-dual field strengths by deriving an effective action in an arbitrary gauge. With the exception of the light cone gauges, this action is in general non-gaussian and, typically, non-local.

Nicolai maps and stochastic observables from a coupling constant flow

Coupling constant flow equations are derived for supersymmetric scalar and gauge theories. Thes equations then serve as a starting point for the construction of Nicolai maps. From the fixed-point properties of the latter we establish a large class of stochastic functionals.

Construction of Supersymmetric Theories by Use of the Local Nicolai Mapping

Construction of Supersymmetric Theories by Use of the Local Nicolai Mapping

Stochastic variables in N = 1 supersymmetric Yang-Mills theory

The stochastic structure of N = 1 supersymmetric Yang-Mills theory is rederived by using a previously developed method for the construction of the (nonlocal) Nicolai map. The stochastic variables correspond to the fixed points of this mapping. The relations are derived in a light cone gauge and in general covariant gauges.

Stochastic quantization, supersymmetry and the Nicolai map

We present a formulation of the method of stochastic quantization of Parisi and W that reveals its intimate connection with supersymmetry. The crucial ingredient of this analysis is the Nicolai map. By using supersymmetric Ward identities, we derive relations between two Fokker-Planck-type Hamiltonians which arise naturally in this formalism.

On the stochastic structure of globally supersymmetric field theories

We reformulate the bosonic sector of globally supersymmetric field theoriesthrough a “fermionisation” of bosonic Feynman graphs. The recipe for the fermionisation gives an explicit realisation of the Nicolai map. The graphical rules for supersymmetric Yang-Mills fields in the reformulated version turn out to be simpler than those of ordinary Yang-Mills fields.